Fuzzy Neural Network for Clustering and Classification

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IJEDR1503031 International Journal of Engineering Development and Research (www.ijedr.org)

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Fuzzy Neural Network for Clustering and

Classification

1_{Archana R. Shinde,}2_{Prof .D.B Kshirsagar} 1_Student,2_Professor,

1_{Department Computer Engineering,}

1_{S.R.E.S. College of Engineering, Kopargaon,Maharashtra India.}

________________________________________________________________________________________________________ Abstract - This paper presents the implementation of Fuzzy Neural Network (FNN) for clustering and classification. Fuzzy neural network combines the advantage of both fuzzy logic and neural network. This paper mainly focuses on implementation of two algorithms. First algorithm is General Fuzzy min max Neural Network (GFMM) training algorithm which combines the processing of supervised and unsupervised data in a single training algorithm. Second algorithm is Data Core Based Fuzzy min max Neural Network (DCFMN) which considers the characteristics of the data and influence of noise simultaneously. These two algorithms provide high accuracy, flexibility, better performance, strong robustness in classification and clustering.

Index Terms - Classification, Clustering, Fuzzy systems, Fuzzy min-max neural networks ,Data core, Overlapped neuron, Pattern classification, Hyperboxes, Hyperbox expansion, Hyperbox contraction, Overlapping neurons, Classifying neurons ,Membership function, Robustness.

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I.INTRODUCTION

Now a days fuzzy logic and neural network are greatly used to develop intelligent systems .The main reason to combine fuzzy logic and neural network is that fuzzy logic have greater ability to handle approximate or uncertain information with the capability of neural networks in learning from processes. This paper presents implementation of two algorithms. One is General Fuzzy min-max Neural Network (GFMM)[4] and second is Data Core Based Fuzzy min-min-max Neural Network (DCFMN)[7].

General Fuzzy min-max Neural Network (GFMM)[4] processes supervised and unsupervised data in a single training algorithm. In labeled or supervised learning ,which is also referred as a pattern classification problem, labels are provided with input patterns. In unlabeled or unsupervised learning, also called as a cluster analysis problem, the input pattern is unlabeled and we must deal with the splitting a set of input patterns into a number of more or less homogenous clusters with respect to a suitable similar property. Patterns which are similar are allocated to the same cluster, while the patterns which differ are put in different clusters.

In many industrial processes, the measurement data may include noise caused by various kinds of processes which affects system performance and may decrease the accuracy of classification. Therefore, Data Core Based Fuzzy min max Neural Network (DCFMN) [7] algorithm is implemented to improve the performance of classification.

II.RELATED WORK

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succession of hyperbox cuts. The generalization capability of ARC/PARC technique mostly depends on the adopted cutting strategy. By using a recursive cutting procedure (RARC and R-PARC) it is possible to obtain better results.

III.SYSTEM ARCHITECTURE

This paper presents the implementation of a fuzzy neural network for clustering and classification. In this fuzzy neural network two training algorithm are implemented for clustering and classification. First algorithm is General Fuzzy min max Neural Network (GFMM)[4] and second is Data Core Based Fuzzy min max Neural Network (DCFMN)[7]. In GFMM single training algorithm is required to processes labeled and unlabeled input patterns.Inthis algorithm learning process is completed in a few passes through the data and of placing and adjusting the hyperboxes in the pattern space which is referred as an expansion–contraction process. To retaining all the interesting features, a number of modifications to their definition have been made in order to accommodate fuzzy input patterns in the form of lower and upper bounds, combine the labeled and unlabeled input patterns, and improve the effectiveness of system , but in GFMM new data can be included without retraining of data. DCFMN considers the characteristics of the data and influence of noise simultaneously while performing classification. In DCFMN , membership function has been designed based on the geometric center and data core in a hyperbox. Instead of using the contraction process, overlapped neuron with new membership function based on the data core is proposed and added to neural network to represent the overlapping area of hyperboxes belonging to different classes. DCFMN has strong robustness and high accuracy in classification taking onto account the effect of data core and noise. The general architecture of the proposed system is shown in figure.1

Figure.1 General System Architecture

A.General Fuzzy min max neural network (GFMM) Algorithm

GFMM algorithm consist of mainly four steps which are Initialization, Expansion, Overlap Test, and Contraction. Basic Terms :

First we see some basic terms that are used in this algorithm 1. Input:

Input provided to the GFMM algorithm is in the form of ordered pair as

{𝑋ℎ,𝑑ℎ}

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where 𝑋_ℎ is the ℎ𝑡ℎ_{input pattern and 𝑑}

ℎ ∈ {1,2,…,𝑝} is the index of one of the 𝑝+1 classes, where 𝑑ℎ =0 means that the input pattern is unlabeled.

2. Fuzzy Hyperbox Membership Function:

Fuzzy Hyperbox Membership Function is very important in fuzzy min max neural network algorithm. The decision whether the given input pattern belongs to a particular class or cluster, thus whether the corresponding hyperbox is to be expanded, depends mainly on the membership value describing the degree to which an input pattern fits within the hyperbox .In GFMM algorithm the membership function of 𝑗𝑡ℎ_{hyperbox for ℎ}𝑡ℎ_{input pattern is calculated as}

𝑏_𝑗(𝑋_ℎ)=min

𝑖=1..,𝑛(min⁡([1 − 𝑓(𝑥ℎ𝑖

𝑢 _{− 𝑤}

𝑗𝑖, 𝛾𝑖)],

[1 − 𝑓(𝑣𝑗𝑖− 𝑥ℎ𝑖𝑙 , 𝛾𝑖)] )) (2) Where 𝑓(𝑟, 𝛾)={

1⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓⁡𝑟𝛾 > 1 𝑟𝛾⁡⁡⁡⁡𝑖𝑓⁡0 ≤ 𝑟𝛾 ≤ 1⁡

0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓⁡𝑟𝛾 < 0 Where 𝑥_ℎ𝑖𝑙 ⁡and 𝑥_{ℎ𝑖⁡⁡}𝑢 is the lower and upper limit of⁡ℎ𝑡ℎ_{input pattern, 𝑣}

𝑗𝑖 and 𝑤𝑗𝑖 is the min and max points of 𝑗𝑡ℎ hyperbox and threshold function; 𝛾=[ 𝛾1,𝛾2,…,𝛾𝑛]—sensitivity parameters which regulates how fast the membership values decrease.

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1. Initialization : Initially min point 𝑉𝑗 and max point 𝑊𝑗 of 𝑗𝑡ℎ_{hyperbox set to}

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑉_𝑗= 0⁡⁡𝑎𝑛𝑑⁡⁡⁡𝑊_𝑗= 0 (3) When first input pattern is presented 𝑗𝑡ℎ_{hyperbox adjusted and min and max points set to}

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑉𝑗⁡ = ⁡ 𝑥ℎ𝑙⁡𝑎𝑛𝑑⁡⁡⁡𝑊𝑗= 𝑥ℎ⁡⁡𝑢 (4)

2. Hyperbox Expansion : When the input pattern is presented, find the hyperbox with the highest degree of membership and allowing expansion (if needed) is expanded to include input pattern . Expansion criteria is

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡∀_i=1…n(max(w_ji, x_hiu_{) − min⁡(v}

ji, xhil )) ≤ θ (5) And

if⁡d_h= 0⁡then⁡adjust⁡B_j else

if⁡class(B_j) =

{

0⁡⁡⁡⁡ ⇒ adjust⁡Bj⁡ class(B_j) = d_h dh⁡⁡⁡ ⇒ ⁡adjust⁡Bj⁡

take else ⇒ another⁡⁡Bj

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We adjust ⁡𝐵_𝑗⁡ in following way.

v_jinew_{= min(v} ji old_{, x}

hil ) ⁡⁡for⁡each⁡i = 1, … , n

w_jinew_{= max(w}

ji old_{, x}

hiu) ⁡⁡for⁡each⁡i = 1, … , n

If none of the existing hyperboxes can expand to include the input pattern 𝑥ℎ , then a new hyperbox 𝐵𝑘 is created, adjusted, and labeled by setting 𝑐𝑙𝑎𝑠𝑠(𝐵_𝑘)=⁡𝑑_ℎ.

The θ is a user-defined parameter that define a bound on the maximum size of a hyperbox and its value significantly affects the effectiveness of the training algorithm.

3. Hyperbox Overlap Test: Due to inclusion of input patterns hyperboxes of different classes are overlapped with each other .So to detect overlapping between hyperboxes overlap test is performed as follows

 If the hyperbox, expanded in the last expansion step ,is not labeled then test for overlapping with all the other hyperboxes. This ensures that all unlabeled hyperboxes do not overlap with any of the other existing ones.

 If the hyperbox 𝐵𝑗 expanded in the last expansion step,belongs to one of the existing classes then test for the overlap only with the hyperboxes not being part of the same class as 𝐵_𝑗. Notice that this allows to overlap the hyperboxes belonging to the same class.

The full hyperbox overlap test can be therefore summarized as follows.

class(Bj) = {

0⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⇒ ⁡test⁡overlapping⁡with⁡ 𝑎𝑙𝑙⁡𝑜𝑡ℎ𝑒𝑟⁡ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑥𝑒𝑠 𝑒𝑙𝑠𝑒 ⇒ ⁡𝑡𝑒𝑠𝑡⁡𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑝𝑖𝑛𝑔⁡𝑜𝑛𝑙𝑦⁡𝑖𝑓

𝑐𝑙𝑎𝑠𝑠(𝐵_𝑗) ≠ 𝑐𝑙𝑎𝑠𝑠(𝐵_𝑘)

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The four cases are being considered (where initially 𝜕𝑜𝑙𝑑_=1).

Case 1:𝑣_𝑗𝑖< 𝑣_𝑘𝑖< 𝑤_𝑗𝑖< 𝑤_𝑘𝑖

⁡⁡𝜕𝑛𝑒𝑤_{= min⁡(𝑤}

𝑗𝑖− 𝑣𝑘𝑖, 𝜕𝑜𝑙𝑑). Case 2:𝑣_𝑘𝑖< 𝑣_𝑗𝑖< 𝑤_𝑘𝑖< 𝑤_𝑗𝑖

⁡𝜕𝑛𝑒𝑤 _{= min⁡(𝑤}

𝑘𝑖− 𝑣𝑗𝑖, 𝜕𝑜𝑙𝑑). Case 3:𝑣_𝑗𝑖< 𝑣_𝑘𝑖≤ 𝑤_𝑘𝑖< 𝑤_𝑗𝑖

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝜕𝑛𝑒𝑤_{= min⁡(min(𝑤}

𝑘𝑖− 𝑣𝑗𝑖, 𝑤𝑗𝑖− 𝑣𝑘𝑖) , 𝜕𝑜𝑙𝑑). Case 4:𝑣_𝑘𝑖< 𝑣_𝑗𝑖≤ 𝑤_𝑗𝑖< 𝑤_𝑘𝑖

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝜕𝑛𝑒𝑤 _{= min⁡(min(𝑤}

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If overlap for the 𝑖𝑡ℎ_{dimension has been detected (one of the above four cases is valid) and}_𝜕𝑜𝑙𝑑_{− 𝜕}𝑛𝑒𝑤 _{> 0⁡, then ∆=i, 𝜕}𝑜𝑙𝑑₌ 𝜕𝑛𝑒𝑤_{and 𝑐𝑎𝑠𝑒 = 𝑙 (𝑙= {1,2,3,4}− the case for which the smallest overlap was found). If overlap for the⁡𝑖}𝑡ℎ_{dimension has not been} detected, set Δ=−1 which indicate that the contraction step is not necessary.

4. Hyperbox Contraction: If Δ>0⁡then only the Δ𝑡ℎ dimensions of the two hyperboxes are adjusted. For contraction again four cases are being considered.

Case 1:𝑣_𝑗∆< 𝑣_𝑘∆< 𝑤_𝑗∆< 𝑤_𝑘∆ 𝑣_𝑘∆𝑛𝑒𝑤= 𝑤_𝑗∆𝑛𝑒𝑤=𝑣𝑘∆

𝑜𝑙𝑑_{+ 𝑤}

𝑗∆𝑜𝑙𝑑 2 or alternatively (𝑤_𝑗∆𝑛𝑒𝑤=𝑣_𝑘∆𝑜𝑙𝑑). Case 2:𝑣𝑘∆< 𝑣𝑗∆< 𝑤𝑘∆< 𝑤𝑗∆

𝑣_𝑗∆𝑛𝑒𝑤_{= 𝑤} 𝑘∆𝑛𝑒𝑤=

𝑣_𝑗∆𝑜𝑙𝑑_{+ 𝑤} 𝑘∆𝑜𝑙𝑑 2 or alternatively (𝑣_𝑗∆𝑛𝑒𝑤_=𝑤

𝑘∆𝑜𝑙𝑑). Case 3:𝑣_𝑗∆< 𝑣_𝑘∆≤ 𝑤_𝑘∆< 𝑤_𝑗∆

𝑖𝑓⁡𝑤𝑘∆− ⁡ 𝑣𝑗∆< 𝑤𝑗∆− 𝑣𝑘∆

𝑡ℎ𝑒𝑛⁡⁡𝑣_𝑗∆𝑛𝑒𝑤 = 𝑤_𝑘∆𝑜𝑙𝑑⁡otherwise (𝑤_𝑗∆𝑛𝑒𝑤=𝑣_𝑘∆𝑛𝑒𝑤) Case 4:𝑣𝑘∆< 𝑣𝑗∆≤ 𝑤𝑗∆< 𝑤𝑘∆

𝑖𝑓⁡𝑤_𝑘∆− ⁡ 𝑣_𝑗∆< 𝑤_𝑗∆− 𝑣_𝑘∆ 𝑡ℎ𝑒𝑛⁡⁡𝑤_𝑘∆𝑛𝑒𝑤_{= 𝑣}

𝑗∆𝑜𝑙𝑑 otherwise (𝑣𝑘∆𝑛𝑒𝑤=𝑤𝑗∆𝑜𝑙𝑑). B.Data Core Based Fuzzy min max Neural Network (DCFMN) Algorithm:

In DCFMN a new membership function for classifying the neuron of DCFMN is defined in which the noise, the geometric center of the hyperbox, and the data core are considered.

1. CN Membership Function: For considering the influence of noise and the density of data in the hyperbox, an improved hyperbox membership function of CN is defined as follows:

𝑏_𝑗(𝑥_ℎ)=min

𝑖=1,..,𝑛(min⁡(𝑓(𝑥ℎ𝑖− 𝑤𝑗𝑖+ 𝜀, 𝑐𝑗𝑖),

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑓(𝑣𝑗𝑖+ 𝜀 − 𝑥ℎ𝑖, 𝑐𝑗𝑖)))) (8) where ε is a parameter representing noise, c is difference between the data core in the hyperbox and the geometric center of the corresponding hyperbox, and f is the ramp threshold function which is defined as.

𝑓(𝑟, 𝑐) = {

𝑒−𝑟2×(1+𝑐)×𝜆_, _{𝑟 > 0, 𝑐 > 0} 𝑒−𝑟2×(1−𝑐)×1/𝜆_, _{𝑟 > 0, 𝑐 < 0}

1,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑟 < 0

2. OLN Membership Function: An OLN is created when there is an overlapping area between two hyperboxes that belong to two different classes 𝑉′_{, ⁡𝑊}′_{nodes in OLN represent the minimum and maximum points of the overlapped area. The activation} function used for OLN is given by

do,q(xh) = g(vo′, wo′, xh, yq)

={ 1 𝑛

0,∑ (1 − |𝑥ℎ𝑖− 𝑦𝑞𝑖|) 𝑛

𝑖=1 ,∀𝑖=1,..𝑛𝑤𝑜𝑖> (𝑥ℎ𝑖) > 𝑣𝑜𝑖 (9)

where o = 1, 2, . . . , l is the index of overlapped node, q =1, 2 is the index of hyperbox which has the same overlap, and⁡⁡𝑦𝑞is the data core of each overlapped hyperbox.

DCFMNLearning Algorithm

DCFMN algorithm consist of three steps as follows:

1. Hyperbox Expansion: Identify the expandable hyperboxes and expand them. For the hyperbox to be expanded, the following expansion criteria must be met:

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𝑣_𝑗𝑖𝑛𝑒𝑤= min(𝑣_𝑗𝑖𝑜𝑙𝑑, 𝑥_ℎ𝑖𝑙 ) ⁡⁡𝑓𝑜𝑟⁡𝑒𝑎𝑐ℎ⁡𝑖 = 1, … , 𝑛

𝑤_𝑗𝑖𝑛𝑒𝑤= max(𝑤_𝑗𝑖𝑜𝑙𝑑, 𝑥_ℎ𝑖𝑢) ⁡⁡𝑓𝑜𝑟⁡𝑒𝑎𝑐ℎ⁡𝑖 = 1, … , 𝑛

2. Hyperbox Overlap Test : Due to inclusion of input patterns two hyperboxes are overlapped with other. So detect overlapping between hyperbox of different classes overlap test is performed. Assuming that 𝛼𝑜𝑙𝑑_{= 1 initially, the four test cases and the} corresponding minimum overlap value for the 𝑖𝑡ℎ_{dimension is as follows.}

Case 1:𝑣𝑗𝑖< 𝑣𝑘𝑖< 𝑤𝑗𝑖< 𝑤𝑘𝑖 𝛼𝑛𝑒𝑤_{= min⁡(𝑤}

𝑗𝑖− 𝑣𝑘𝑖, 𝛼𝑜𝑙𝑑). Case 2:𝑣𝑘𝑖< 𝑣𝑗𝑖< 𝑤𝑘𝑖< 𝑤𝑗𝑖

𝛼𝑛𝑒𝑤_{= min⁡(𝑤}

𝑘𝑖− 𝑣𝑗𝑖, 𝛼𝑜𝑙𝑑). Case 3:𝑣𝑗𝑖< 𝑣𝑘𝑖≤ 𝑤𝑘𝑖< 𝑤𝑗𝑖

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝛼𝑛𝑒𝑤_{= min⁡(min(𝑤}

𝑘𝑖− 𝑣𝑗𝑖, 𝑤𝑗𝑖− 𝑣𝑘𝑖) , 𝛼𝑜𝑙𝑑). Case 4:𝑣_𝑘𝑖< 𝑣_𝑗𝑖≤ 𝑤_𝑗𝑖< 𝑤_𝑘𝑖

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝛼𝑛𝑒𝑤_{= min⁡(min(𝑤}

𝑘𝑖− 𝑣𝑗𝑖, 𝑤𝑗𝑖− 𝑣𝑘𝑖) , 𝛼𝑜𝑙𝑑).

If 𝛼𝑜𝑙𝑑_−𝛼𝑛𝑒𝑤_>_{0, then there is an overlap for the 𝑖}𝑡ℎ_{dimensions, and let ∆}₌_i_{. Otherwise, the test stops and set 𝑖}𝑡ℎ _{= 0, and} nothing is done.

3. Adding OLN - If an overlap between hyperboxes of different classes exists, the process of adding OLN is performed as follows.

Case1:⁡𝑣_𝑗𝑖< 𝑣_𝑘𝑖< 𝑤_𝑗𝑖< 𝑤_𝑘𝑖

𝑣_{𝑚+𝑜𝑖}′ = 𝑣𝑘𝑖⁡⁡, 𝑤𝑚+𝑜𝑖′ = 𝑤𝑗𝑖 Case 2:𝑣𝑘𝑖 < 𝑣𝑗𝑖< 𝑤𝑘𝑖 < 𝑤𝑗𝑖

𝑣_{𝑚+𝑜𝑖}′ _{= 𝑣}

𝑗𝑖⁡⁡, 𝑤𝑚+𝑜𝑖′ = 𝑤𝑘𝑖. Case 3:𝑣𝑗𝑖< 𝑣𝑘𝑖 ≤ 𝑤𝑘𝑖 < 𝑤𝑗𝑖

𝑣_{𝑚+𝑜𝑖}′ _{= 𝑣}

𝑘𝑖⁡⁡, 𝑤𝑚+𝑜𝑖′ = 𝑤𝑘𝑖 Case 4:𝑣𝑘𝑖 < 𝑣𝑗𝑖≤ 𝑤𝑗𝑖< 𝑤𝑘𝑖

𝑣_{𝑚+𝑜𝑖}′ _{= 𝑣}

𝑗𝑖⁡⁡, 𝑤𝑚+𝑜𝑖′ = 𝑤𝑗𝑖

where o = (1, 2, . . . , l) is index of neuron to be added to the network, and 𝑉′_and_𝑊′_{are used to store the minimum and} maximum points of the hyperboxes.

IV.IMPLEMENTATION DETAILS

This system mainly presents implementation of two algorithm, GFMM and DCFMN and our objective is to implements these two algorithm for standard datasets iris and wine .GFMM is four step algorithm which are Initialization, Expansion, Overlap Test, and Contraction which are successfully completed and gives desired output.

Figure 2.shows browsing of dataset as input for our system .Iris dataset is browsed which contains 150 records of three classes .All input patterns are labeled.

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Figure 3. shows that number of hyperboxes created of different classes , here user defined parameter θ set 0.1 and number of hyperboxes created is 139. As we increase value of θ from 0.1 to 0.9 ,number of hyperboxes created is decreases.

Figure 3: Number of Hyperboxes created.

Figure 4. Shows overlapping found between hyperboxes and contraction of hyperboxes performed to remove overlapping of hyperboxes of different classes.

Figure 4: Overlap test and contraction

Up to this step training of algorithm is completed , now testing is performed by presenting input pattern to the system and system classify that input pattern correctly and displaying class of that input pattern which shown in figure 5. Misclassification rate of GFMM algorithm is very less.

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V.CONCLUSION

This paper presents the implementation of fuzzy neural network for clustering and classification which uses two training algorithm for clustering and classification. These two training algorithm is based on fuzzy logic and neural network. First algorithm is General Fuzzy min max Neural Network which combines the processing of supervised and un supervised input data within a single training algorithm and provide better performance. Second algorithm is Data Core Based Fuzzy min max Neural Network which considers the characteristics of the data and influence of noise simultaneously and provide strong robustness against noise and better performance in classification

VI.ACKNOWLEDGMENT

I would like to take this opportunity to express my profound gratitude and deep regard to my Project Guide Prof. D. B. Kshirsagar, for his exemplary guidance, valuable feedback and constant encouragement throughout the duration of the project. His valuable suggestions were of immense help throughout my project work. His perceptive criticism kept me working to make this project in a much better way. Working under him was an extremely knowledgeable experience for me.

REFERENCES

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